It is well-established that the pair \(|{-}| \dashv \operatorname{Sing}\)  is a simplicial Quillen equivalence (see e.g. [1]}).
We have also seen in Theorem REF  that the adjunction \(\tilde{\mathsf {c}} \dashv \operatorname{ev}_*\)  is a simplicial Quillen equivalence.
The commutativity of (REF ) follows from the definitions (REF ) of the functors \(Re\)  and \(S\) , which use \(|{-}|\)  and \(\operatorname{Sing}\) , respectively.
The fact that \(Re \dashv S\)  is a Quillen equivalence then follows from the two-out-of-three property of Quillen equivalences and Corollary REF .
